Integrated Nested Laplace Approximations (INLA) Demonstration

Kenneth A. Flagg

Motivation

Point Processes and Sampling

  • Modeled as (log of) latent Gaussian Process

Point Processes and Sampling

  • Observe points
  • Make inferences about intensity function

Point Processes and Sampling

  • Brix and Møller (2001) one of very few examples

INLA Introduction

Why INLA?

  • Rue, Martino, and Chopin (2009)
  • Bayesian Hierarchical models
    • Many latent Gaussian variables
    • Few parameters
    • E.g. spatial prediction using Gaussian process model
  • Monte Carlo methods impractical

Why INLA?

  • Approximate posterior marginals using Laplace expansions
    • Clever algebra, end up with posterior in denominator
    • Taylor expand log-posterior about its mode
    • Results in a Gaussian approximation
    • Some numerical integration needed

Advantages and Disadvantages

  • Advantages
    • Accurate approximation
    • Fast computation of many posterior marginals
  • Disadvantages
    • Does not provide full joint posterior
    • Slow with >4-6 parameters

Normal Example

Normal Example

Blangiardo and Cameletti (2015) section 4.9

  • \(\mathbf{y} = (y_{1}, \dots, y_{n})'\) independent Gaussian observations
  • \(y_{i} \sim \mathsf{N}(\theta, \sigma^{2})\)
  • \(\theta \sim \mathsf{N}(\mu_{0}, \sigma_{0}^{2})\)
  • \(\psi = 1/\sigma^{2}\), \(\psi \sim \mathrm{Gamma}(a, b)\)

The posterior distribution of the nuisance parameter is

\[p(\psi|\mathbf{y}) \propto \frac{p(\mathbf{y} | \theta, \psi) p(\theta) p(\psi)} {p(\theta | \psi, \mathbf{y})}\]

Normal Example

  • Priors: \(\mu_{0} = -3\), \(\sigma_{0}^{2} = 4\), \(a = 1.6\), \(b = 0.4\)
  • 30 observed points

Normal Example

Normal Example

Normal Example

Normal Example

Spatial Point Pattern: Trees in a Rainforest

Spatial Point Pattern

Spatial Point Pattern

References

References

Blangiardo, Marta, and Michela Cameletti. 2015. Spatial and Spatio-Temporal Bayesian Models with R-INLA. Wiley.

Brix, Anders, and Jesper Møller. 2001. “Space-Time Multi Type Log Gaussian Cox Processes with a View to Modelling Weeds.” Scandinavian Journal of Statistics 28 (3): 471–88.

Rue, Håvard, Sara Martino, and Nicolas Chopin. 2009. “Approximate Bayesian Inference for Latent Gaussian Models by Using Integrated Nested Laplace Approximations.” Journal of the Royal Statistical Society: Series B (Statistical Methodology) 71 (2): 319–92.